{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.5.6"
    },
    "colab": {
      "name": "confidence-intervals.ipynb",
      "provenance": [],
      "include_colab_link": true
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/opi-lab/optical-metrology-2021/blob/gh-pages/notebooks/confidence_intervals.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "JLUFXFfZMIEv"
      },
      "source": [
        "# Illustration of Confidence Intervals"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Fakq773QMIEw"
      },
      "source": [
        "**Example 4.1. Illustration of Confidence Intervals.** \n",
        "Assume a sample of N= 31 independent observations are collected \n",
        "from a normally distributed random variable x with the following results:\n",
        "\n",
        "    60, 61, 47, 56, 61, 63, \n",
        "    65, 69, 54, 59, 43, 61, \n",
        "    55, 61, 56, 48, 67, 65, \n",
        "    60, 58, 57, 62, 57, 58, \n",
        "    53, 59, 58, 61, 67, 62, 54\n",
        "\n",
        "Determine a 90% confidence interval for the mean value and variance of the random variable x."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "yMYYmlojMIEw"
      },
      "source": [
        "## Solution\n",
        "Let's go to chapter 4 (Statistical Principals), section 4 (**Confidence Intervals**) of J. Bendat, A Piersol."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "scrolled": true,
        "id": "qMULZFmzMIEx",
        "outputId": "42068872-fd08-41ec-a43c-d67e1314b986"
      },
      "source": [
        "from IPython.display import IFrame\n",
        "IFrame('pdfs/J-Bendat--A-Piersol-random-data-analysis-and-measurement-procedures-section-4-4.pdf',\n",
        "       width='100%', height=400)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/html": [
              "\n",
              "        <iframe\n",
              "            width=\"100%\"\n",
              "            height=\"400\"\n",
              "            src=\"pdfs/J-Bendat--A-Piersol-random-data-analysis-and-measurement-procedures-section-4-4.pdf\"\n",
              "            frameborder=\"0\"\n",
              "            allowfullscreen\n",
              "        ></iframe>\n",
              "        "
            ],
            "text/plain": [
              "<IPython.lib.display.IFrame at 0x10da97550>"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 5
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 336
        },
        "id": "Z977f6_GMIEy",
        "outputId": "d3c7b638-0e58-4cc1-eb01-dc1c29149ce8"
      },
      "source": [
        "import numpy as np\n",
        "from scipy import stats\n",
        "import seaborn as sns\n",
        "%matplotlib inline\n",
        "\n",
        "# The data\n",
        "X = np.array([60, 61, 47, 56, 61, 63, 65, 69, 54, 59, 43, 61, 55, 61,\n",
        "              56, 48, 67, 65, 60, 58, 57, 62, 57, 58, 53, 59, 58, 61, 67, 62, 54])\n",
        "\n",
        "# Let's visualize the distribution\n",
        "sns.distplot(X)"
      ],
      "execution_count": 1,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "/usr/local/lib/python3.7/dist-packages/seaborn/distributions.py:2557: FutureWarning: `distplot` is a deprecated function and will be removed in a future version. Please adapt your code to use either `displot` (a figure-level function with similar flexibility) or `histplot` (an axes-level function for histograms).\n",
            "  warnings.warn(msg, FutureWarning)\n"
          ],
          "name": "stderr"
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.axes._subplots.AxesSubplot at 0x7f72761f6ad0>"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 1
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": [],
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "a9MLchnTMIEz",
        "outputId": "b1ce9490-a7b3-4e2b-d82f-4beba28526d2"
      },
      "source": [
        "# Number of data\n",
        "N = float(len(X))\n",
        "\n",
        "# We estimate the sample mean\n",
        "mu = X.mean()\n",
        "\n",
        "# We estimate the sample variance and standar deviation with N-1 degrees of freedom\n",
        "s2 = X.var(ddof=1)\n",
        "s = X.std(ddof=1)\n",
        "\n",
        "# We print the values\n",
        "print('mean = %4.2f' % mu)\n",
        "print('s2 = %4.2f' % s2)\n",
        "print('s = %4.2f' % s)\n",
        "\n",
        "\n",
        "# We need to determine confidence intervals with 90% confidence\n",
        "# 90% confidence means an alpha = 0.1\n",
        "# The confidence intervals for μ_x are given by Eq. (4.46b)\n",
        "\n",
        "# We need the t-student statistic for t(1-alpha/2,N-1)\n",
        "t = stats.t.ppf(1-0.1/2,N-1) # ppf: percent point function (inverse of cdf)\n",
        "\n",
        "# The confidence interval\n",
        "conf_int_mu = (mu-t*s/np.sqrt(N),mu+t*s/np.sqrt(N))\n",
        "\n",
        "print('confidence interval for μ_x: (%2.2f < %2.2f < %2.2f)' % (conf_int_mu[0],mu,conf_int_mu[1]))\n",
        "# Another way to do it using stats.t.interval \n",
        "print('confidence interval for μ_x: (%2.2f, %2.2f)' % stats.t.interval(1-0.1, N-1, loc=mu, scale=s/np.sqrt(N-1)))"
      ],
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "mean = 58.61\n",
            "s2 = 33.45\n",
            "s = 5.78\n",
            "confidence interval for μ_x: (56.85 < 58.61 < 60.38)\n",
            "confidence interval for μ_x: (56.82, 60.40)\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "kTxJrBmEMIE0"
      },
      "source": [
        "## Additional work \n",
        "1. Compute the 90% confidence interval for the variance $s^2_x$. Use Eq. (4.47) and the function [``stats.chi2.ppf()``.](http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.chi2.html#scipy.stats.chi2 \"scipy.stats.chi2 — SciPy v0.17.0 Reference Guide\")\n",
        "2. Compute the 95% confidence intervals for the mean $\\mu_x$ and variance $s^2_x$."
      ]
    }
  ]
}